10
76
(K10a
73
)
A knot diagram
1
Linearized knot diagam
6 9 10 7 8 1 5 4 3 2
Solving Sequence
2,9
3 10
4,6
1 7 8 5
c
2
c
9
c
3
c
1
c
6
c
8
c
5
c
4
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
+ 4u
7
+ u
6
− 5u
5
− 3u
4
+ 2u
2
+ b + 3u + 1, −u
6
+ 3u
4
− 2u
2
+ a − 1,
u
11
− u
10
− 5u
9
+ 4u
8
+ 9u
7
− 4u
6
− 5u
5
− 3u
4
− 3u
3
+ 5u
2
+ 3u + 1i
I
u
2
= hu
11
− 3u
9
− u
8
+ 2u
7
+ 2u
6
+ 3u
5
− 3u
3
− 2u
2
+ b − u, 2u
17
− 12u
15
+ ··· + a + 3, u
18
− u
17
+ ··· + 2u − 1i
I
u
3
= hb, a + 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 30 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h−u
9
+4u
7
+· · · +b + 1, −u
6
+3u
4
−2u
2
+a − 1, u
11
−u
10
+· · · +3u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
10
=
−u
−u
3
+ u
a
4
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
6
− 3u
4
+ 2u
2
+ 1
u
9
− 4u
7
− u
6
+ 5u
5
+ 3u
4
− 2u
2
− 3u − 1
a
1
=
u
3
− 2u
−u
3
+ u
a
7
=
u
3
− 2u
−u
10
+ u
9
+ 4u
8
− 3u
7
− 5u
6
+ 2u
5
+ u
3
+ 3u
2
a
8
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
4
+ u
2
+ 1
u
9
− u
8
− 4u
7
+ 2u
6
+ 5u
5
+ u
4
− 3u
2
− 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
10
+ 6u
9
+ 8u
8
− 26u
7
− 14u
6
+ 34u
5
+ 16u
4
+ 4u
3
− 10u
2
− 30u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
11
+ 3u
10
+ 6u
9
+ 7u
8
+ 7u
7
+ 3u
6
− 2u
5
− 8u
4
− 7u
3
− 5u
2
− 2u − 2
c
2
, c
3
, c
4
c
5
, c
7
, c
9
u
11
− u
10
− 5u
9
+ 4u
8
+ 9u
7
− 4u
6
− 5u
5
− 3u
4
− 3u
3
+ 5u
2
+ 3u + 1
c
8
, c
10
u
11
+ 3u
10
+ ··· − 16u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
11
+ 3y
10
+ ··· − 16y −4
c
2
, c
3
, c
4
c
5
, c
7
, c
9
y
11
− 11y
10
+ ··· − y −1
c
8
, c
10
y
11
+ 7y
10
+ ··· + 24y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.062122 + 0.811051I
a = −1.82009 − 0.72518I
b = 0.762686 + 0.875309I
4.60381 + 2.87937I −1.58714 − 3.23335I
u = −0.062122 − 0.811051I
a = −1.82009 + 0.72518I
b = 0.762686 − 0.875309I
4.60381 − 2.87937I −1.58714 + 3.23335I
u = −1.32132
a = 0.669088
b = −0.992754
−6.97991 −12.6670
u = −1.296720 + 0.321683I
a = −0.591796 + 0.578733I
b = 0.958422 − 0.661375I
−3.08453 + 5.20915I −9.44226 − 3.72118I
u = −1.296720 − 0.321683I
a = −0.591796 − 0.578733I
b = 0.958422 + 0.661375I
−3.08453 − 5.20915I −9.44226 + 3.72118I
u = 1.360100 + 0.374662I
a = −1.56319 − 0.53861I
b = 0.764438 − 1.080520I
−4.40916 − 11.51290I −10.44081 + 7.44023I
u = 1.360100 − 0.374662I
a = −1.56319 + 0.53861I
b = 0.764438 + 1.080520I
−4.40916 + 11.51290I −10.44081 − 7.44023I
u = 1.42406 + 0.13076I
a = 0.601423 + 0.717547I
b = −0.273627 + 1.210650I
−11.39950 − 4.33574I −15.3124 + 3.6840I
u = 1.42406 − 0.13076I
a = 0.601423 − 0.717547I
b = −0.273627 − 1.210650I
−11.39950 + 4.33574I −15.3124 − 3.6840I
u = −0.264651 + 0.295634I
a = 1.039110 − 0.325568I
b = −0.215541 − 0.601634I
−0.314917 + 0.927579I −5.88395 − 7.40073I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.264651 − 0.295634I
a = 1.039110 + 0.325568I
b = −0.215541 + 0.601634I
−0.314917 − 0.927579I −5.88395 + 7.40073I
6
II.
I
u
2
= hu
11
−3u
9
+· · ·+b −u, 2u
17
−12u
15
+· · ·+a +3, u
18
−u
17
+· · ·+2u −1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
10
=
−u
−u
3
+ u
a
4
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
−2u
17
+ 12u
15
+ ··· − 2u − 3
−u
11
+ 3u
9
+ u
8
− 2u
7
− 2u
6
− 3u
5
+ 3u
3
+ 2u
2
+ u
a
1
=
u
3
− 2u
−u
3
+ u
a
7
=
−2u
17
+ 12u
15
+ ··· − 2u − 3
u
14
− 4u
12
+ ··· + 3u
2
+ u
a
8
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−2u
17
+ 12u
15
+ ··· − 3u − 2
u
13
− 5u
11
− 2u
10
+ 9u
9
+ 8u
8
− 4u
7
− 10u
6
− 6u
5
+ 5u
3
+ 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
17
− 24u
15
− 4u
14
+ 56u
13
+ 20u
12
− 48u
11
− 36u
10
− 24u
9
+
16u
8
+ 64u
7
+ 24u
6
− 12u
5
− 20u
4
− 24u
3
− 8u
2
− 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
2
c
2
, c
3
, c
4
c
5
, c
7
, c
9
u
18
− u
17
+ ··· + 2u − 1
c
8
, c
10
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
2
c
2
, c
3
, c
4
c
5
, c
7
, c
9
y
18
− 13y
17
+ ··· − 12y + 1
c
8
, c
10
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.11181
a = −0.294140
b = 0.512358
−2.09142 −3.34770
u = −0.138557 + 0.857281I
a = 1.70857 + 0.83690I
b = −0.728966 − 0.986295I
0.30826 + 7.08493I −6.42320 − 5.91335I
u = −0.138557 − 0.857281I
a = 1.70857 − 0.83690I
b = −0.728966 + 0.986295I
0.30826 − 7.08493I −6.42320 + 5.91335I
u = −1.112360 + 0.436175I
a = −0.238783 + 0.723669I
b = 0.628449 − 0.875112I
−2.67293 − 2.45442I −9.67208 + 2.91298I
u = −1.112360 − 0.436175I
a = −0.238783 − 0.723669I
b = 0.628449 + 0.875112I
−2.67293 + 2.45442I −9.67208 − 2.91298I
u = −0.535620 + 0.576021I
a = −0.792096 − 0.581161I
b = 0.140343 + 0.966856I
−5.07330 + 2.09337I −12.51499 − 4.16283I
u = −0.535620 − 0.576021I
a = −0.792096 + 0.581161I
b = 0.140343 − 0.966856I
−5.07330 − 2.09337I −12.51499 + 4.16283I
u = 0.035822 + 0.749326I
a = 1.96913 + 0.59401I
b = −0.796005 − 0.733148I
1.08148 − 1.33617I −4.71591 + 0.70175I
u = 0.035822 − 0.749326I
a = 1.96913 − 0.59401I
b = −0.796005 + 0.733148I
1.08148 + 1.33617I −4.71591 − 0.70175I
u = −1.209730 + 0.357771I
a = 0.429481 − 0.621272I
b = −0.796005 + 0.733148I
1.08148 + 1.33617I −4.71591 − 0.70175I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.209730 − 0.357771I
a = 0.429481 + 0.621272I
b = −0.796005 − 0.733148I
1.08148 − 1.33617I −4.71591 + 0.70175I
u = 1.253840 + 0.303492I
a = −1.61989 − 0.98839I
b = 0.628449 − 0.875112I
−2.67293 − 2.45442I −9.67208 + 2.91298I
u = 1.253840 − 0.303492I
a = −1.61989 + 0.98839I
b = 0.628449 + 0.875112I
−2.67293 + 2.45442I −9.67208 − 2.91298I
u = 1.308540 + 0.065670I
a = −0.41325 − 1.38121I
b = 0.140343 − 0.966856I
−5.07330 − 2.09337I −12.51499 + 4.16283I
u = 1.308540 − 0.065670I
a = −0.41325 + 1.38121I
b = 0.140343 + 0.966856I
−5.07330 + 2.09337I −12.51499 − 4.16283I
u = 1.311030 + 0.356898I
a = 1.61494 + 0.70203I
b = −0.728966 + 0.986295I
0.30826 − 7.08493I −6.42320 + 5.91335I
u = 1.311030 − 0.356898I
a = 1.61494 − 0.70203I
b = −0.728966 − 0.986295I
0.30826 + 7.08493I −6.42320 − 5.91335I
u = 0.285873
a = −3.02207
b = 0.512358
−2.09142 −3.34770
11
III. I
u
3
= hb, a + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
−1
a
3
=
1
1
a
10
=
1
0
a
4
=
0
1
a
6
=
−1
0
a
1
=
1
0
a
7
=
−1
0
a
8
=
0
−1
a
5
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
c
10
u
c
2
, c
3
, c
7
u + 1
c
4
, c
5
, c
9
u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
c
10
y
c
2
, c
3
, c
4
c
5
, c
7
, c
9
y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
2
· (u
11
+ 3u
10
+ 6u
9
+ 7u
8
+ 7u
7
+ 3u
6
− 2u
5
− 8u
4
− 7u
3
− 5u
2
− 2u − 2)
c
2
, c
3
, c
7
(u + 1)
· (u
11
− u
10
− 5u
9
+ 4u
8
+ 9u
7
− 4u
6
− 5u
5
− 3u
4
− 3u
3
+ 5u
2
+ 3u + 1)
· (u
18
− u
17
+ ··· + 2u − 1)
c
4
, c
5
, c
9
(u − 1)
· (u
11
− u
10
− 5u
9
+ 4u
8
+ 9u
7
− 4u
6
− 5u
5
− 3u
4
− 3u
3
+ 5u
2
+ 3u + 1)
· (u
18
− u
17
+ ··· + 2u − 1)
c
8
, c
10
u(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
2
· (u
11
+ 3u
10
+ ··· − 16u − 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
2
· (y
11
+ 3y
10
+ ··· − 16y −4)
c
2
, c
3
, c
4
c
5
, c
7
, c
9
(y −1)(y
11
− 11y
10
+ ··· − y −1)(y
18
− 13y
17
+ ··· − 12y + 1)
c
8
, c
10
y(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
2
· (y
11
+ 7y
10
+ ··· + 24y −16)
17
